The theory that forbade flight

Newton’s Principia contains a theory of fluid resistance, and it makes a specific prediction about an inclined plate: the force goes as the square of the angle of incidence.

That single exponent did more to delay powered flight than any material shortage. A wing large enough to lift a man, at any sensible angle, needs an engine of a size nobody could build if the lift goes as $\sin^2\alpha$. The arithmetic was done repeatedly through the nineteenth century, it always came out the same way, and it was wrong.

The theory, stated fairly #

Treat the air as a stream of independent particles. Each one that strikes the surface gives up the component of its momentum normal to that surface, keeps the tangential component, and slides away. Nothing interacts with anything else, and a particle that would have missed the plate is unaffected by it.

The pressure follows in two lines. The mass flux striking unit area of a surface inclined at $\theta$ to the stream is $\rho U \sin\theta$, and each unit of that mass gives up a normal velocity $U\sin\theta$. So the pressure is $\rho U^2 \sin^2\theta$, or

$$C_p = 2\sin^2\theta$$

and for a flat plate at incidence $\alpha$, resolving into lift and drag,

$$C_L = 2\sin^2\alpha\cos\alpha, \qquad C_D = 2\sin^3\alpha$$

Two factors of the sine, one from how much air arrives and one from how much each bit of it is turned. That is the whole of the model and it is not a foolish one — it is the correct answer for a genuinely collisionless gas, and it is exactly how one would compute the force on a satellite in the upper atmosphere.

What is wrong with it #

The momentum argument is not the error. A wing does hold itself up by turning air downwards, and the momentum theorem done properly gives the right answer.

The error is the assumption that only the air that strikes the surface is involved. In a real continuum flow the pressure field extends far above and below the wing, and air that never comes near the surface is deflected all the same — because the fluid is continuous, and pushing on one parcel pushes on its neighbours. The mass of air a wing turns per second is many times the mass flux through its own frontal area, and Newton’s model has no mechanism for any of it.

The consequence is a completely different power law. When the mass affected is set by the span rather than by the projected area, the deflected mass no longer contains a factor of $\sin\alpha$, and one of the two sines disappears: the lift becomes linear in the angle, which is the 2πα of thin-aerofoil theory.

The exponent is a statement about how much air is involved, and that is exactly the quantity the impact picture gets wrong.

The size of the error is π over the angle #

The two theories can be divided, and the answer is worth committing to memory.

For small $\alpha$, Newton gives $C_L \approx 2\alpha^2$ and thin-aerofoil theory gives $2\pi\alpha$, so

$$\frac{C_L^{\text{Newton}}}{C_L^{\text{real}}} \approx \frac{2\alpha^2}{2\pi\alpha} = \frac{\alpha}{\pi}$$

At five degrees that is $0.0873/\pi = 0.028$, the factor of thirty-six above. At two degrees it is one part in ninety. The error diverges as the angle shrinks, which is the worst possible behaviour for a theory of flight, because small angles are where flight happens.

The same ratio says where the two would agree: $\alpha = \pi$ radians, which is meaningless, so the honest statement is that they never agree at any incidence a wing uses. What actually happens at large angles is that the real wing stalls and its lift curve bends over towards — but not onto — the impact result.

Where the theory becomes exact #

Now the reversal, and it is the reason this essay is worth writing rather than merely noting.

At hypersonic speeds the shock wave produced by a body lies extremely close to its surface. The air is not deflected far in advance; it is turned abruptly, in a thin layer, at the surface itself. That is precisely the geometry Newton’s model assumes, and the impact formula becomes the right answer to a very good approximation — the Newtonian approximation, which is the standard first estimate for hypersonic pressure distributions and is still used to lay out re-entry shapes.

The site can test that claim with its own machinery rather than asserting it. The exact pressure behind an oblique shock at wave angle $\beta$ is

$$C_p = \frac{4\left(M^2\sin^2\beta - 1\right)}{(\gamma + 1)M^2} \;\xrightarrow{\ M\to\infty\ }\; \frac{4\sin^2\beta}{\gamma + 1}$$

which is Newton’s $2\sin^2\beta$ multiplied by $2/(\gamma+1)$.

Newton was right about a gas nobody breathes #

The remaining factor $2/(\gamma+1)$ is a property of the gas rather than of the geometry, and it goes to one as $\gamma \to 1$.

$\gamma = 1$ describes a gas with so many internal degrees of freedom that compressing it raises its temperature not at all — every scrap of the compression work goes into internal modes. That is not an ordinary gas at ordinary temperature. It is, however, a fair description of air behind a very strong shock, where the molecules are vibrating, dissociating and ionising, and the effective $\gamma$ falls towards 1.1 or below.

So the theory that forbade flight is exact for a re-entering capsule, where the air behind the bow shock is doing all of those things at once.

What the solver computed, and how it was checked #

Three assertions, and each catches a different way of being wrong.

The limit is the limit. At $M = 10^4$ the exact oblique-shock pressure divided by Newton’s value is $0.83333333$ against $2/(\gamma+1) = 0.83333333$, at four different wave angles.

The limit closes at $\gamma \to 1$. Repeating the same computation with $\gamma = 1 + 10^{-9}$ gives a ratio of $1.00000000$, which is the statement that the two theories coincide there and not merely nearby.

It has to be hopeless at ordinary incidence. The check requires Newton’s lift coefficient to be under a tenth of the thin-aerofoil value at five degrees — it is 2.8 per cent — and requires the Newtonian lift to be quadratic, by comparing one degree with two. A version of the formula that had been “corrected” into agreement with reality at small angles would fail both, which is the point: the theory has to stay wrong where it is wrong.

What it says about drag, which is worse #

The lift was wrong by a factor of thirty-six; the drag is wrong in a way that is qualitatively different and is worth a sentence.

Newton’s model gives $C_D = 2\sin^3\alpha$, which vanishes at zero incidence. That is correct for this model — a plate edge-on to a hail of particles intercepts nothing — and it is a peculiar kind of right answer, because a real plate at zero incidence has drag from friction on both sides and the impact model contains no friction at all. Newton’s fluid has no viscosity and no tangential force whatever.

So the theory predicts, for a streamlined body at zero incidence, exactly zero drag — which is the same conclusion the exact inviscid theory reaches by a completely different route, and is wrong for entirely different reasons. Two models, two mechanisms, one shared blind spot: neither has a boundary layer, so neither has the drag that a smooth body at zero incidence actually pays.

Why the mistake was so durable #

Two reasons, and they are worth separating.

It is dimensionally reasonable and qualitatively right. The force increases with angle, increases with speed squared, increases with area, and points roughly where it should. Everything about it is plausible except the exponent, and an exponent is exactly the sort of thing an argument from plausibility cannot check.

It was tested against the wrong experiments. Eighteenth and nineteenth century measurements of resistance were mostly on bluff bodies and flat plates at large angles, where $\sin^2\alpha$ is not far wrong — the two theories differ most at small incidence, which is the case nobody was measuring because nobody was flying.

Otto Lilienthal’s careful measurements in the 1890s, and then the Wright brothers’ wind-tunnel work in 1901, are where the small-angle behaviour was finally pinned down. The Wrights’ immediate problem was not Newton but Smeaton’s coefficient — a similarly venerable constant, wrong by about 40 per cent, which they measured for themselves and corrected before building anything.

The sine-squared picture in the wild #

Two places where the impact formula genuinely applies are worth naming, because they show what the model needs to be true.

A satellite in the upper atmosphere. Above about 150 km the mean free path is longer than the spacecraft, so molecules genuinely do arrive independently, strike, and leave without interacting with each other. That is Newton’s assumption exactly, and the drag on a satellite is computed with formulae of this shape — corrected for whether the molecules re-emit specularly or diffusely, which is the one thing the model has to be told.

A re-entry capsule. Here it is not the mean free path but the geometry: the shock lies on the body, and the turning is local. The Newtonian estimate gives the pressure distribution over a blunt capsule well enough to size a heat shield with, and it is the standard first pass.

The common condition is that the deflected air must be confined to the surface — either because there is no continuum to spread the influence, or because a shock has pinned it there. Every ordinary aerodynamic case fails that condition, which is why the theory fails for aircraft and works for spacecraft. It is the clearest example on this site of a model whose validity is a statement about a regime rather than about the mathematics.

A third case is worth a caution rather than an endorsement. A flat plate at very large incidence — a sail aback, a stalled wing, a barn door — is sometimes said to obey the sine-squared law, and it does not: the separated flow behind it has a base pressure of its own that the impact model knows nothing about, and the drag of a bluff body is dominated by that pressure rather than by the momentum arriving on the front.

What the molecules do after they arrive #

Newton’s model follows each particle up to the surface and then makes an assumption about what happens next: it gives up its normal momentum, keeps its tangential momentum, and slides away. That second half is a physical claim, it is the source of the theory’s zero tangential force, and in the one regime where the model is supposed to be exact it is wrong.

A molecule striking a real surface is not reflected like a billiard ball. It is adsorbed, spends a moment on the surface, and is re-emitted in a direction with no memory of the one it came from — a cosine distribution about the surface normal, with a speed set by the wall’s temperature rather than by its own arrival speed. That is diffuse re-emission, and engineering surfaces in low orbit are close to fully diffuse, largely because they are covered in adsorbed atomic oxygen.

Two consequences follow, and both contradict the impact picture in the regime it owns.

There is a tangential force at zero incidence. A molecule arriving with tangential momentum and leaving with none on average has transferred all of it to the surface. So a plate edge-on to a free-molecular stream feels a drag, where Newton’s model gives exactly zero — the same quantity the essay noted above as the theory’s peculiar blind spot, and it turns out not to need a boundary layer to appear.

And the force depends on the temperature of the paint. The re-emitted molecules leave carrying momentum set by the surface temperature, so the total force on a satellite depends on how hot its surface is — an aerodynamic force with a thermal parameter in it, and no fluid-mechanical analogue anywhere else in this collection.

The practical size of it is a standing difficulty in orbital work. A compact satellite’s drag coefficient is usually taken as about 2.2 rather than the 2 the pure impact model gives, and the difference is entirely the re-emission. Since it is not measured for any particular spacecraft, it is one of the largest uncertainties in predicting where a satellite will be next week and when it will come down.

So the theory is exact about the geometry and approximate about the surface. Newton assumed a collisionless gas correctly and a wall incorrectly — which is the same division of labour the rest of this collection keeps finding, arriving from an unexpected direction.

How to tell the two apart in one measurement #

The two theories differ in shape rather than in size, so a single test settles them and does not need absolute accuracy.

Measure the lift at two incidences and take the ratio. Newton’s theory predicts a ratio of four between two degrees and one; thin-aerofoil theory predicts two. No calibration is needed, no reference area, no density, no tunnel correction — everything that is hard to measure cancels in the ratio, and the two predictions differ by a factor of two.

That is worth stating because it is the general shape of a good test of a model. What is easy to measure badly is a level; what is easy to measure well is a ratio or a slope, and a theory that can be refuted by a slope can be refuted by an experimenter with a poor balance and a good protractor. Lilienthal, working with a whirling arm and no tunnel at all, had the slope right in the 1890s.

The same principle runs through this site’s own checks: the ring’s speed is asserted through the slope of its logarithm rather than through its value, because the slope is the part that survives the modelling choices.

What the picture cannot show #

No real body is a flat plate of zero thickness. The Newtonian estimate applied to a real hypersonic shape uses the local surface inclination everywhere and includes a shadowed region where $C_p = 0$, which this essay’s two-dimensional plate does not draw.

The centrifugal correction is missing. A curved hypersonic surface has fluid moving along it in a thin layer, and the pressure needed to turn that layer adds a term to the Newtonian value — the Busemann correction — which is what turns the estimate from a first approximation into a usable one.

Nothing here is a real gas. The whole $\gamma \to 1$ argument is about a calorically perfect gas with a lower $\gamma$, which is a caricature of what dissociating air actually does. Real hypersonic calculations carry chemistry, and this site has none.

Where the model stops #

The impact theory has no continuum in it at all, which is why it is exactly right in the two limits where the continuum is irrelevant: free-molecular flow, where particles genuinely do not interact, and hypersonic flow, where the shock layer is so thin that the deflection is effectively local. In between — which is all of aeronautics up to Mach 5 or so — it is wrong, and it is wrong by more the smaller the angle gets.

The thin-aerofoil result it is being compared with has its own limits, and they are the opposite ones: 2πα holds for small angles and fails at large ones, where a real wing stalls and Newton’s curve begins, coincidentally, to look better again. Two theories, each exact in the regime the other cannot reach, and neither of them any use in the other’s.

Who found it, and when #

Newton’s resistance theory is in Book II of the Principia, 1687, and the $\sin^2$ result is Proposition 34’s corollary. He was careful about what he was assuming — a rare medium of separated particles — and the assumption was carried forward far beyond its warrant by others.

The correction is essentially Lanchester’s and Kutta’s and Joukowski’s, between 1894 and 1906, and it required the idea of circulation rather than any better bookkeeping about impacts. The rehabilitation came in the 1950s, when Lees and others found that Newton’s formula was an excellent hypersonic estimate, and the Newtonian approximation entered the design of every re-entry vehicle since.

Where the ladder goes next #

One misconception down, and the next is not about lift at all but about a word. The upper surface of a wing is universally described as being under suction, which implies that the air there is pulling on it — and a fluid cannot pull.